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Register a new userQuasi-Gasdynamic Approach for Numerical Solution of Magnetohydrodynamic Equations
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We introduce an application of the Quasi-Gasdynamic method for a solution of ideal magnetohydrodynamic equations in the modeling of compressible conductive gas flows. A time-averaging procedure is applied for all physical parameters in order to obtain the quasi-gas-dynamic system of equations for magnetohydrodynamics. Evolution of all physical variables is presented in an unsplit divergence form. Divergence-free evolution of the magnetic field is provided by using a constrained transport method based on Stokes theorem. Accuracy and convergence of this method are verified on a large set of standard 1D and 2D test cases.
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In this paper, we investigate numerically a diffuse interface model for the Navier-Stokes equation with fluid-fluid interface when the fluids have different densities cite{Lowengrub1998}. Under minor reformulation of the system, we show that there is a continuous energy law underlying the system, assuming that all variables have reasonable regularities. It is shown in the literature that an energy law preserving method will perform better for multiphase problems. Thus for the reformulated system, we design a $C^0$ finite element method and a special temporal scheme where the energy law is preserved at the discrete level. Such a discrete energy law (almost the same as the continuous energy law) for this variable density two-phase flow model has never been established before with $C^0$ finite element. A Newtons method is introduced to linearise the highly non-linear system of our discretization scheme. Some numerical experiments are carried out using the adaptive mesh to investigate the scenario of coalescing and rising drops with differing density ratio. The snapshots for the evolution of the interface together with the adaptive mesh at different times are presented to show that the evolution, including the break-up/pinch-off of the drop, can be handled smoothly by our numerical scheme. The discrete energy functional for the system is examined to show that the energy law at the discrete level is preserved by our scheme.
Fractional variational approach has gained much attention in recent years. There are famous fractional derivatives such as Caputo derivative, Riesz derivative and Riemann-Liouville derivative. Sever
For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with a {it Dirichlet} boundary condition imposed along the disk, Neugebauer and Meinel in the 1990s famously derived an explicit expression for the spacetime metric corresponding to the Bardeen-Wagoner uniformly rotating disk of dust. In this paper, we consider a similar boundary value problem for a rotating disk in which a {it Neumann} boundary condition is imposed along the disk instead of a Dirichlet condition. Using the integrable structure of the Ernst equation, we are able to reduce the problem to a Riemann-Hilbert problem on a genus one Riemann surface. By solving this Riemann-Hilbert problem in terms of theta functions, we obtain an explicit expression for the Ernst potential. Finally, a Riemann surface degeneration argument leads to an expression for the associated spacetime metric.
We present a new dynamical proof of the Thouless-Anderson-Palmer (TAP) equations for the classical Sherrington-Kirkpatrick spin glass at sufficiently high temperature. In our derivation, the TAP equations are a simple consequence of the decay of the two point correlation functions. The methods can also be used to establish the decay of higher order correlation functions. We illustrate this by proving a suitable decay bound on the three point functions from which we derive an analogue of the TAP equations for the two point functions.
An overview is given on recent developments in the affine Weyl group approach to Painleve equations and discrete Painleve equations, based on the joint work with Y. Yamada and K. Kajiwara.
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